I'm always pretty nervous before interviews, but so far in my career (also software engineering) I've never not been offered the job after the final interview, even when I've stuttered pretty badly all the way through. It's nice because I think it reinforces the fact that people are genuinely only interested in your skillset and whether you'd be a good addition to their team personality wise. I always thought "but surely if an employer has got the choice of two people with the same skills, but one has a stutter and the other hasn't, obviously they'll choose the person without one", but even if that were true, realistically two different people never have exactly the same skillset and personality, so it's a non issue. Congratulations on the new job!